On one-sided primitivity of Banach algebras

Details On one-sided primitivity of Banach algebras On one-sided primitivity of Banach algebras

2008-07-31

arxiv.org/abs/0807.5033v2

Mathematics

Functional Analysis

14 pages. To appear, with minor changes, in the Proceedings of the Edinburgh Mathematical Society

Scientific paper

Let $S$ be the semigroup with identity, generated by $x$ and $y$, subject to $y$ being invertible and $yx=xy^2$. We study two Banach algebra completions of the semigroup algebra $\mathbb{C}S$. Both completions are shown to be left-primitive and have separating families of irreducible infinite-dimensional right modules. As an appendix, we offer an alternative proof that $\mathbb{C}S$ is left-primitive but not right-primitive. We show further that, in contrast to the completions, every irreducible right module for $\mathbb{C}S$ is finite dimensional and hence that $\mathbb{C}S$ has a separating family of such modules.

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